Global topology of the Hitchin system
Tamas Hausel
TL;DR
This paper surveys the complex interplay of ideas from physics, geometry, and number theory in understanding the cohomology of the Hitchin system's moduli space, highlighting recent results and conjectures.
Contribution
It provides a comprehensive overview of the current state of research and conjectures on the cohomology of the Hitchin system, integrating diverse mathematical and physical perspectives.
Findings
Connections between gauge theory and mirror symmetry in Hitchin systems
Conjectures relating cohomology to Langlands duality
Insights into the interplay of geometry, physics, and number theory
Abstract
Here we survey several results and conjectures on the cohomology of the total space of the Hitchin system: the moduli space of semi-stable rank n and degree d Higgs bundles on a complex algebraic curve C. The picture emerging is a dynamic mixture of ideas originating in theoretical physics such as gauge theory and mirror symmetry, Weil conjectures in arithmetic algebraic geometry, representation theory of finite groups of Lie type and Langlands duality in number theory.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Algebra and Geometry · Homotopy and Cohomology in Algebraic Topology